Mathematicians Dan Goldston and Cem Yildirim announced an important new result in late March 2003, which while not proving that there are infinitely many twin primes does substantially advance the understanding of "Small Gaps Between Primes", as their paper is called. It is said* to be one of the most important advances in prime theory for many decades.

On his homepage Goldston describes his current research as follows:

I have been working since 1985 on methods for proving that there are arbitrarily large primes that are unusually close together. The goal is to prove that for p and p' primes the infimum of the ratio (p-p')/ log p is zero. Here log p is the average distance between primes around p, and therefore we are trying to find consecutive primes within any fixed proportion of the average spacing. The current best result of Maier from 1986 shows that this ratio is infinitely often less than 0.248. Cem Yildirim and I are currently writting a series of papers on Higher Correlations of Short Divisor Sums which we hope will provide new tools both for small gaps between primes and other problems involving primes.

Early work by Hardy and Littlewood in the 1920s proved that if the General Riemann Hypothesis is true, then p' - p is infinitely often less than (2/3) log p. Later authors removed the dependence on unproved hypotheses and reduced the constant 2/3 to 0.248. What Goldston and Yildirim have now shown is that that for any ε > 0, there are infinitely many p with p' - p < ε log p.

The primary argument of this paper is that a sieve can be constructed which isolates the twin primes from the rest of the natural numbers, and this sieve can further adhere to a property described as the LeapfrogLemma. In short, the purpose of this lemma is to encapsulate the Odd-Even Theorem and other properties of sieves in such a way so that an induction or other proof of infinity can be established, and then any sieve which has all the properties matching those found in this lemma is known to produce an infiniteset of whatever object is under scrutiny.

There is some speculation on a method for transforming the work in the above paper in order to apply the concepts of the main lemma to a wider selection of sieves. One of the current contenders is to change the function definitions so that the functions accept a vectorinput and return a set of values as the output. This particular approach could allow non-single-function sieves to be evaluated using the lemma, but brings about the question of whether sieves can be combined and retain valued properties, and if so, what limitations on this process might exist.